Solve the equation. Do not use a calculator. log ⁡ 6 8 x = log ⁡ 6 48 \log _ { 6 } 8 x = \log _ { 6 } 48 lo g 6 8 x = lo g 6 48 A) x = 48 x = 48 x = 48 B) x = 6 x = 6 x = 6 C) x = 1 x = 1 x = 1 D) x = 8 x = 8 x = 8 E) x = 7 x = 7 x = 7

Question

Solve the equation. Do not use a calculator.   log ⁡ 6 8 x = log ⁡ 6 48 \log _ { 6 } 8 x = \log _ { 6 } 48 lo g 6 ​ 8 x = lo g 6 ​ 48 A)    x = 48 x = 48 x = 48 B)    x = 6 x = 6 x = 6 C)    x = 1 x = 1 x = 1 D)    x = 8 x = 8 x = 8 E)    x = 7 x = 7 x = 7

Kyl jane Vasquez · Accepted Answer

Final Answer : B, Explanation :   Using the fact that   log ⁡ b ( b n ) = n \log _ { b } ( b ^ { n } ) = n lo g b ​ ( b n ) = n  , we can simplify the equation as follows:   log ⁡ 6 8 x = log ⁡ 6 48 ⇒ log ⁡ 6 ( 8 x / 48 ) = log ⁡ 6 ( 1 / 6 ) \log _ { 6 } 8 x = \log _ { 6 } 48 \Rightarrow \log _ { 6 } (8x/48) = \log _ { 6 } (1/6) lo g 6 ​ 8 x = lo g 6 ​ 48 ⇒ lo g 6 ​ ( 8 x /48 ) = lo g 6 ​ ( 1/6 ) Now use the definition of logarithm    log ⁡ b c =    ⟺    b d = c \log _ { b } c=\iff b^{d} = c lo g b ​ c = ⟺ b d = c   with base 6 6 log ⁡ 6 ( 8 x / 48 ) = 6 log ⁡ 6 ( 1 / 6 ) ⇒ 8 x / 48 = 1 / 6 6^{\log _ { 6 } (8x/48)} = 6^{\log _ { 6 } (1/6)} \Rightarrow 8x/48 = 1/6 6 l o g 6 ​ ( 8 x /48 ) = 6 l o g 6 ​ ( 1/6 ) ⇒ 8 x /48 = 1/6 Solving the above equation, we get   x = 6 x=6 x = 6  . Therefore, the answer is B)    x = 6 x = 6 x = 6  .

Solve the equation. Do not use a calculator. $log _ { 6 } 8 x = \log _ { 6 } 48$

A) $x = 48$
B) $x = 6$
C) $x = 1$
D) $x = 8$
E) $x = 7$

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Solve the equation. Do not use a calculator. log⁡68x=log⁡648\log _ { 6 } 8 x = \log _ { 6 } 48log6​8x=log6​48 A) x=48x = 48x=48 B) x=6x = 6x=6 C) x=1x = 1x=1 D) x=8x = 8x=8 E) x=7x = 7x=7

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Solve the equation. Do not use a calculator. $log _ { 6 } 8 x = \log _ { 6 } 48$

A) $x = 48$
B) $x = 6$
C) $x = 1$
D) $x = 8$
E) $x = 7$